Chapter 10: Advanced hypercontractivity

In this chapter we complete the proof of the Hypercontractivity Theorem for uniform $\pm 1$ bits. We then generalize the $(p,2)$ and $(2,q)$ statements to the setting of arbitrary product probability spaces, proving the following:

We can thereby extend all the consequences of the basic Hypercontractivity Theorem for $f : \{-1,1\}^n \to {\mathbb R}$ to functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, except with quantitatively worse parameters depending on “$\lambda$”. We also introduce the technique of randomization/symmetrization and show how it can sometimes eliminate this dependence on $\lambda$. For example, it’s used to prove Bourgain’s Sharp Threshold Theorem, a characterization of boolean-valued $f \in L^2(\Omega^n, \pi^{\otimes n})$ with low total influence which has no dependence at all on $\pi$.